Almost sure finiteness for the total occupation time of an (d,α,β)-superprocess
Xiaowen Zhou, Concordia University
Abstract
For $0<α≤ 2$ and $0<β≤ 1$ let $X$ be the $(d,α,β)$-superprocess,
i.e. the superprocess with $α$-stable spatial movement in $bR^d$ and
$(1+β)$-stable branching. Given that the initial measure $X_0$ is Lebesgue
on $bR^d$, Iscoe conjectured in [7] that the total occupational time $int_0^infty
X_t(B)dt$ is a.s. finite if and only if $dβ<α$, where $B$ denotes any
bounded Borel set in $reels^d$ with non-empty interior.
In this note we give a partial answer to Iscoe's conjecture by showing that
$int_0^infty X_t(B)dt<infty$ a.s. if $2dβ<α$ and, on the other hand,
$int_0^infty X_t(B)dt=infty$ a.s. if $dβ> α$.
For $2dβ<α$, our result can also imply the a.s. finiteness of the total
occupation time (over any bounded Borel set) and the a.s. local extinction for
the empirical measure process of the $(d,al,β)$-branching particle system
with Lebesgue initial intensity measure.
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